CST – Computer Simulation Technology


Electrostatic Field Between Two Charged Spheres

Calculated potential distribution in the vicinity of two charged spheres. Figure 1: Potential distribution of two charged conducting spheres.

The Physics

The investigated setup consists of two conducting, charged spheres with radius R0=1m in free space. In classic textbooks this setup is often used to present an enhanced example for the method of imaging, see e.g. [1].

The electrostatic problem (no time dependency) consists of the two PEC spheres; the midpoints are separated by C=R0/0.2 and the potential difference between both spheres is defined as U=1V. Following [1], the highest electric field strength Emax can be found at the points on the spheres which are closest to each other (P1 and P2 in Fig. 2). The expected, analytical result is Emax0.74U. Please note that the exact value depends on the truncation of the infinite series.

Diagram of the problem with all relevant variables Figure 2: Diagram of the problem with all relevant variables labeled, according to [1].

The Model

The model can be easily constructed and simulated with CST EM STUDIO, the low frequency and statics module of CST STUDIO SUITE.

The CAD modeler has a predefined object ‘Sphere’ and so the setup is quickly done. The option of a potential definition can be found in the simulation ribbon. By choosing open boundary conditions, appropriate symmetry planes and a tetrahedral mesh with curved elements, the simulation can be fast and accurately performed. In order to improve the accuracy of the simulation, a vacuum sphere is defined around each PEC sphere, with a radius 0.1m greater than that of the PEC sphere. This produces a denser mesh around the sphere, where the gradient of the field is expected to be greatest.

Model of the problem Figure 3: Problem setup in CST STUDIO SUITE. All relevant distances are shown.
Parameter Value Description
R0 1 m PEC sphere radius
distance 5 m Seperation of sphere midpoints
U 1 V Potential between spheres

In order to extract the field value at a certain point in the 3 dimensional calculation domain, the 3d post processing template "Evaluate field in arbitrary coordinates" is used.

Model Construction Watch Video

Download Model File (Student Edition) Download Model File

Discussion of Results

calculated absolute value of the electric field
Figure 4: Field plot, showing the absolute value (left) and vector representation (right) of the E-field

The provided images show the results in a 2D cut-plane. From the absolute field pattern, it is clear that the highest electric field is at the inner faces of the two spheres. The vector plot helps to understand the field distribution. The arrows are always perpendicular to the PEC surfaces, indicating that the field is normal to the surface of the spheres, and the open boundaries of the simulation setup have no visual influence on the direction and strength of the fields close to the boundaries of the calculation domain.

By varying the distance between spheres, the maximum field strength can plotted vs the distance as shown in Figure 5. It can be seen that the analytical value Emax=0.74V/m is met even with a quite coarse mesh. Separating the spheres further decreases the field strength – bringing them closer together leads to higher field values.

Field strength at the point P2 Figure 5: Calculated electric field strength (absolute value) at the point P2


Will the electric field introduce a force on the spheres? If so, in which direction does it point? Use the "Forces" tool under the post processing ribbon to calculate the force acting on the spheres relative to the origin.


  • [1] K. Küpfmüller, W. Mathis, A. Reibiger, Theoretische Elektrotechnik: Eine Einführung, 18. Auflage, Springer-Lehrbuch, pp. 141-143
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